Problem: Find the gradient of $f(x, y) = 2xy + \sin(x)$. $\nabla f = ($ $,$ $)$
Solution: The gradient of a scalar field is all its partial derivatives put together into a vector. For a 2D scalar field, this looks like $\nabla f = (f_x, f_y)$. Let's find $f_x$ and $f_y$. $\begin{aligned} f_x &= \dfrac{\partial}{\partial x} \left[ 2xy + \sin(x) \right] \\ \\ &= 2y + \cos(x) \\ \\ f_y &= \dfrac{\partial}{\partial y} \left[ 2xy + \sin(x) \right] \\ \\ &= 2x \end{aligned}$ The gradient of $f$ is $\nabla f = (2y + \cos(x), 2x)$.